A nice application mentioned in [1]} says that if \(f:M\rightarrow \mathbb {R}\)  is a Morse–Bott function with non-degenerate critical submanifold \(C\subseteq M\)  then there exists an Euler-like vector field \(X\)  for \((M,C)\)  such that \(\mathcal {L}_Xf=2f\)  near \(C\) . Therefore, in the resulting tubular neighborhood embedding \(\phi \)  associated to \(X\)  we have \(\phi ^*X = \mathcal {E}\) , thus obtaining that
\(\mathcal {L}_{\mathcal {E}}\phi ^\ast f = 2\phi ^\ast f\) . This means that \(\phi ^*f\)  is homogeneous of degree 2 and hence
coincides with its quadratic approximation \(c+Q_f\)  (here \(c\)  is the common value of \(f\)  on \(C\) ). As a consequence, we have obtained that \(\phi ^*f=c+Q_f\)  which is just the Morse–Bott lemma.
